Question

How do you find all local maximum and minimum points using the second derivative test given `y=(x+5)^(1/4)`?

Answer 1

`(x+5)^(1/4)` is a monotonically increasing function and there are no maxima or minima.

Explanation:

We observe that the domain of `x` is `[-5,oo)`

As `y=(x+5)^(1/4)`

`y'=1/4(x+5)^((1-1/4))=1/4(x+5)^(-3/4)=1/(4(x+5)^(3/4))`

It is apparent that `x+5` is always positive within the domain `[-5,oo)`

Further `y''=1/4xx-3/4xx(x+5)^(-7/4)=(-3)/(16(x+5)^(7/4)` and is always negative and hence

`(x+5)^(1/4)` is a monotonically increasing function and there are no maxima or minima.
graph{(x+5)^(1/4) [-8.71, 11.29, -3.64, 6.36]}